We attempt the use of a unitary operator to approximate the lattice Boltzmann collision operator. We use a modified amplitude encoding to bypass the renormalization that would have required classical processing at every step (thus eroding any quantum advantage to be had). We describe the hard-wiring of the lattice Boltzmann symmetries into the quantum circuit and show that, for the specific case of the cavity flow, approximating the nonlinear system is limited to low velocities. These findings may help us understand better the possibilities of nonlinear simulations on a quantum computer, and also pave the way for a discussion on how quantum machine learning might be harnessed to address more complex problems.
We attempt the use of a unitary operator to approximate the lattice Boltzmann collision operator. We use a modified amplitude encoding to bypass the renormalization that would have required classical processing at every step (thus eroding any quantum advantage to be had). We describe the hard-wiring of the lattice Boltzmann symmetries into the quantum circuit and show that, for the specific case of the cavity flow, approximating the nonlinear system is limited to low velocities. These findings may help us understand better the possibilities of nonlinear simulations on a quantum computer, and also pave the way for a discussion on how quantum machine learning might be harnessed to address more complex problems.
This study investigates the application of quantum machine learning (QML) to approximate the nonlinear component of the collision operator within the quantum lattice Boltzmann method (QLBM). To achieve this, we train a variational quantum circuit (VQC) to construct an operator U. When applied to the post-linear-collision quantum state ∣Ψi⟩, this operator yields a final state ∣Ψf⟩=U∣Ψi⟩ that successfully replicates the nonlinear collision dynamics derived from the Bhatnagar-Gross-Krook (BGK) approximation. Within this framework, we present two distinct architectures: the R1 model and the R2 model. The R1 model is designed for quantum simulations that involve multiple time steps without intermediate measurements, focusing on accurately capturing nonlinear dynamics in continuous evolution. In contrast, the R2 model is tailored to achieve the high-precision reconstruction of the nonlinear operator for a single time step with an unitary operator.
We present a full quantum algorithm for the lattice Boltzmann method for simulating fluid flows, the only such algorithm to implement both the streaming and collision substeps as unitary operators using an efficient number of qubits. We use Hamiltonian simulation as the main route underlying the algorithm, and show that it uses qubits that scale logarithmically in the flow Reynolds number but gates that scale only polynomially. We motivate the discussion by a brief overview of existing attempts at simulating classical fluids on quantum computers and present a pedagogical discussion on assigning quantum operators to classical variables in both the streaming and the collision substeps, after highlighting the incompatibility of the latter with the implementations of the streaming step that exists in the literature. We use the Bhatnagar–Gross–Krook ansatz for the collision term, representing the relaxation toward an equilibrium distribution. For nonlinear collisions, we use Kowalski's framework that links the nonlinear dynamics of a system to the evolution of bosonic modes, assigning a Carleman linearization order to the truncation in the Fock space of the bosons. We present the qubit and gate complexities, in terms of the chosen accuracy and the Reynolds number. In the Appendix, we work out the details of implementing the operators of the truncated bosonic Fock space in terms of single-qubit gates as well as the error scaling for a general polynomial driving function.
Quantum computing (QC) has emerged as a potential computational tool in fluid dynamics. However, a practical QC protocol with end-to-end practical applications to fluid flow is still in nascent. We here propose a simple and practical hybrid QC algorithm based on lattice Boltzmann methods (LBM), which we term HQLBM, for fluid dynamics simulations. The existing QC-based LBMs are limited to simple low-Reynolds-number flow simulations due to their disability to handle the nonlinear collision term in LBM for QC. In this work, a linearized non-equilibrium collision, derived from kinetic theory, is presented for HQLBM simulating practical flows. The present approach ensures the unitary of quantum algorithms while keeping the collision relaxation parameter adjustable for simulating different Reynolds number flows. Most importantly, the linearized collision keeps the accuracy of the original nonlinear one in the lattice Boltzmann equation when it theoretically recovers its macroscopic counterpart—Navier–Stokes equations. The practicality of the proposed method is demonstrated by simulating two typical flows, including lid-driven and natural convection flows in a square cavity at different Reynolds and Rayleigh numbers, respectively. It is shown that the proposed HQLBM has good numerical accuracy compared to the conventional LBM, and lower complexity on comparisons to the QC-based LBM carried out on single-circuit QC even when it is performed on a Qiskit virtual quantum computer. This work offers a practical application of QC-based LBM for complex fluid dynamics problems.
The Lattice Boltzmann method (LBM) is a well-established mesoscopic approach for simulating fluid dynamics by evolving particle distribution functions on discrete lattices. While the LBM is highly parallelizable on classical hardware, its translation to quantum algorithms is impeded by the collision process, which is intrinsically nonlinear and irreversible. Several existing quantum formulations implement this process through repeated quantum tomography and state preparation at every timestep, leading to significant overheads. We introduce a quantum LBM based on a denoising-type collision operator that avoids tomography-based updates. The collision dynamics are reformulated as an orthogonal projection onto the linearized manifold of equilibrium distributions around a reference state. This geometric approach filters non-equilibrium components while preserving lattice symmetries and approximating nonlinear terms needed to recover hydrodynamic behavior. A complete pipeline is presented with efficient gate-level realizations, incorporating encoding of distributions, collision, streaming, boundary conditions, and measurement of physical quantities such as hydrodynamic forces. In addition, we outline an approach for implementing projector-based operators deterministically without postselection, paving the way to fully coherent multi-timestep LBM simulations. Numerical experiments for advection-diffusion and flow problems demonstrate that the method reproduces macroscopic behaviors with high accuracy, with performance depending on the choice of reference state.
Quantum computing holds transformative potential for simulating nonlinear physical systems, such as fluid turbulence. However, mapping nonlinear dynamics to the linear operations required by quantum hardware remains a fundamental challenge. Here, we bridge this gap by introducing a novel node-level ensemble description of a lattice gas, which enables the simulation of nonlinear fluid dynamics on quantum computers. This approach combines the advantages of the lattice Boltzmann method with low-dimensional representation (computational cost) and lattice gas cellular automata with linear collision treatment (quantum compatibility). Building on this framework, we propose a quantum lattice Boltzmann method that relies on linear operations with medium dimensionality, offering the potential for quantum speedup. We validated the algorithm through simulations of vortex-pair merging and decaying turbulence on up to 16.8 million computational grid points. The results demonstrate remarkable agreement with direct numerical simulation, effectively capturing the essential nonlinear mechanisms of fluid dynamics. This work potentially advances the development of quantum algorithms for other nonlinear problems across various transport phenomena in engineering.
We propose a quantum algorithm for the Lattice Boltzmann (LB) method to simulate fluid flows in the low Reynolds number regime. First, we encode the particle distribution functions (PDFs) as probability amplitudes of the quantum state and demonstrate the need to control the state of the ancilla qubit during the initial state preparation. Second, we express the LB algorithm as a matrix-vector product by neglecting the quadratic non-linearity in the equilibrium distribution function, wherein the vector represents the PDFs, and the matrix represents the collision and streaming operators. Third, we employ classical singular value decomposition (SVD) to decompose the non-unitary collision and streaming operators into a product of unitary matrices. Finally, we show the importance of having a Hadamard gate between the collision and the streaming operations. Our approach has been tested on linear/linearized flow problems such as the advection-diffusion of a Gaussian hill, Poiseuille flow, Couette flow, and lid-driven cavity problems. We provide counts for two-qubit controlled-NOT (CNOT) and single-qubit U gates for test cases involving 9 to 12 qubits, with grid sizes ranging from 24 to 216 points. While the gate count aligns closely with theoretical limits, the high number of two-qubit gates on the order of 107 necessitates careful attention to circuit synthesis.